Cuboctahedron

A polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is one of the Archimedean solids and more particularly, one of the quasi-regular polyhedra.

A cuboctahedron has octahedral symmetry, and its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either. The canonical coordinates for the vertices of a cuboctahedron centered at the origin are (±1,±1,0), (±1,0,±1), (0,±1,±1). Its dual polyhedron is the rhombic dodecahedron[?].

Cuboctahedra are important in spherical close packings. Each sphere can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close packed lattice they correspond to the corners of an anticuboctahedron, formed by twisting a cuboctahedron about one of the four equatorial planes that intersect six vertices. The two halves that each of these planes split the cuboctahedron into are called triangular cupolae, so the anticuboctahedron is also called a triangular orthobicupola. Each of these are Johnson solids.

There are distortions of the cuboctahedron with tetrahedral symmetry that, while no longer edge uniform, are still vertex uniform. These are analogous to the rhombicuboctahedron and rhombicosidodecahedron[?], and can be made by cutting the edges off a tetrahedron and trimming the resulting hexagonal faces. Cuboctahedra and octahedra together make up one of the Andreini tesselations.